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My 86 FXRs was like that. Most of the time it worked when you turned on the key and it was in neutral. Sometimes it didn't work until you started it. So, I always made sure, if the light wasn't on, I was on the bike and had the clutch pulled in. It never gave me a false neutral indication. It was one of those deals that I never really thought about until your post....
Mine did it when new- was a loose connection atone of the wires on the neutral switch located on top of the tranny. Service guy took the wires off, squeezed the connectorcaps w/pliers, putthem back on, & problem solved.
I agree with the others....the connection is a weak source. Check that first! I bet that solves your problem....while you are there....I suggest putting just a little dilectric grease in the connection. It helps keeps moisture away and corrosion away.
If that does not cure the problem....the light bulb socket in the fairing are cheasy. They are very small and are fragile. Vibration and age can effect the connection at the bulbs. Try this last though....because it is a pain to get there.
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